Difference between revisions of "Renormalization Group"
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| − | The | + | The [[Renormalization Group]] approach is a strategy for dealing with problems |
involving many length scales. The strategy is to tackle the problem in steps, one | involving many length scales. The strategy is to tackle the problem in steps, one | ||
| − | step for each length scale. In the case of critical phenomena, the problem, | + | step for each length scale. In the case of critical phenomena, the problem, technically, is to carry out statistical averages over thermal fluctuations on all size scales. The renormalization group approach is to integrate out the fluctuations |
| − | scales. The renormalization group approach is to integrate out the fluctuations | ||
in sequence starting with fluctuations on an atomic scale and then moving to | in sequence starting with fluctuations on an atomic scale and then moving to | ||
successively larger scales until fluctuations on all scales have been averaged out, | successively larger scales until fluctuations on all scales have been averaged out, | ||
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point. Below the Curie temperature, an ideal ferromagnet exhibits spontaneous | point. Below the Curie temperature, an ideal ferromagnet exhibits spontaneous | ||
magnetization in the absence of’ an external magnetic field; the direction of’ the | magnetization in the absence of’ an external magnetic field; the direction of’ the | ||
| − | magnetization depends on the history of the magnet. Above the Curie | + | magnetization depends on the history of the magnet. Above the Curie temperature Tc, there is no spontaneous magnetization.<ref>Ken Wilson, ''Nobel Prize Lecture'' (2018-06) https://www.nobelprize.org/uploads/2018/06/wilson-lecture-2.pdf</ref> |
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| − | <ref>Ken Wilson, ''Nobel Prize Lecture'' (2018-06) https://www.nobelprize.org/uploads/2018/06/wilson-lecture-2.pdf</ref> | ||
Revision as of 22:29, 21 June 2024
Full Title or Meme
Renormalization Groups are used to solve a difficult class of problems where fluctuations persist out to macroscopic wavelengths, and fluctuations on all intermediate length scales are important but cannot be solved by averages.
Solution
The Renormalization Group approach is a strategy for dealing with problems involving many length scales. The strategy is to tackle the problem in steps, one step for each length scale. In the case of critical phenomena, the problem, technically, is to carry out statistical averages over thermal fluctuations on all size scales. The renormalization group approach is to integrate out the fluctuations in sequence starting with fluctuations on an atomic scale and then moving to successively larger scales until fluctuations on all scales have been averaged out, To illustrate the renormalization group ideas the case of’ critical phenomena will be discussed in more detail. First the mean field theory of Landau will be described, and important questions defined. The renormalization group will be presented as an improvement to Landau’s theory.[1] The Curie point of a ferromagnet will be used as a specific example of a critical point. Below the Curie temperature, an ideal ferromagnet exhibits spontaneous magnetization in the absence of’ an external magnetic field; the direction of’ the
magnetization depends on the history of the magnet. Above the Curie temperature Tc, there is no spontaneous magnetization.[2]- ↑ Landau and Lifshitz Statistical Physics (1958) Addison Wesley Chapter XIV
- ↑ Ken Wilson, Nobel Prize Lecture (2018-06) https://www.nobelprize.org/uploads/2018/06/wilson-lecture-2.pdf
